Improvement of test in contingency tables based on asymptotic approximation.

1. 考虑r × s列联表为全多项式模型，在独立零假设下，导出了基于Φ-divergence的检验统计量分布的渐近展开式。表达式由连续项和不连续项组成。同时考虑了连续项和不连续项的取值。利用表达式中假设连续分布的渐近展开式项，构造了在独立假设2下提高统计量分布的卡方近似的小样本精度的变换。将r × s列联表看作乘积多项式模型，在所有多项式总体齐次的零假设下，导出了检验统计量分布的幂散度渐近展开式。表达式由连续项和不连续项组成。当每个总体的样本数量相同时，还考虑了连续项和不连续项的评估。在全多项式模型的独立性检验中，通过假设随机变量的连续性，分别在局部替代和非局部替代假设下，导出了基于幂散度的检验统计量分布的渐近展开式。

1. In the case that r X s contingency table is considered as full multinomial model, we derive expression of asymptotic expansion for the distribution of the test statistics based on Φ-divergence under independent null hypothesis. The expression consists of continuous and discontinuous terms. Evaluation for the continuous and discontinuous term is also considered. Using the term of asymptotic expansion assuming a continuous distribution in the expression, we construct transformation for improving small-sample accuracy of chi-square approximation of the distribution of the statistic under independent hypothesis.2. In the case that r X s contingency table is considered as product multinomial model, we derive expression of asymptotic expansion for the distribution of the test statistics based on power-divergence under null hypothesis that all of the multinomial populations are homogeneity. The expression consists of continuous and discontinuous terms. Evaluation for the continuous and discontinuous term is also considered when the number of samples from each population is the same.3. In the test of independence in full multinomial model, by assuming the continuity of the random variables, we derive asymptotic expansions for the distribution of the test statistics based on power-divergence under local alternative and non-local alternative hypotheses, respectively.